1
1 Electrodynamics of solids — Introduction to practicum
Contents
1 Electrodynamics of solids — Introduction to practicum 1
1.1 Dielectric function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fresnel coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Models of the dielectric function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.1 Lorentz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.2 Drude model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.3 Sellmeier and Cauchy models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.4 Complex Lorentzian model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.5 Gaussian lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.6 Tauc-Lorentz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.7 Critical point parabolic band . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.8 Bruggeman effective medium approximation and surface roughness . . . . . . . 6
1.4 Principles of ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Available spectroscopic equippment . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Practicum tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5.1 Phonon spectra of isotropic and anisotropic semiconductors . . . . . . . . . . . 9
1.5.2 Direct interband transitions in semiconductors . . . . . . . . . . . . . . . . . . 9
1.5.3 Free charge carrier response in semiconductors and metals . . . . . . . . . . . . 9
In this introduction, some of the theoretical approaches used in the analysis of optical spectra
of solids are briefly summarized. For more details, readers are encouraged to consults the following
books, see Refs. [1–3].
1.1 Dielectric function
The goal of the optical spectroscopy is to determine the real and imaginary part of the dielectric
function ε(ω) = ε1(ω) + iε2(ω) that is defined as a ratio of the polarization (volume density of the
electric dipoles) P and the average total electric field E
ε(ω) = 1 +
P(ω)
ε0E(ω)
, (1)
where P(ω) and E(ω) are the Fourier coefficients of the corresponding time dependent quantities.
Dielectric function is connected to the index of refraction as N(ω) = ε(ω). It is often useful, particularly
for conducting materials, to express the optical constants in terms of the optical conductivity
σ(ω) = −iω 0(ε(ω) − 1), where 0 is the permittivity of vacuum. The real part of the optical conductivity
σ1(ω) represents the absorption of electromagnetic radiation per unit of frequency and at zero
frequency it yields the DC conductivity.
1.2 Fresnel coefficients
Consider an electromagnetic wave incident from a material with index of refraction Ni on a planar
interface with another material with the index of refraction Nt. The ratio of the electric field of
reflected wave (r) or transmitted wave (t) with respect to the incident wave is given by the so-called
Fresnel coefficients. In case of isotropic materials, the coefficients read
rs =
Ni cos θi − Nt cos θt
Ni cos θi + Nt cos θt
, rp =
Nt cos θi − Ni cos θt
Nt cos θi + Ni cos θt
, (2)
2
ts =
2Ni cos θi
Ni cos θi + Nt cos θt
, tp =
2Ni cos θi − Ni cos θt
Nt cos θi + Ni cos θt
, (3)
where θi and θt are the angles of the incident and transmitted wave with respect to normal. The
indexes ‘s’ and ‘p’ denote the wave with perpendicular and parallel electric vector with respect to the
plane of incidence. In case of the near normal incidence reflectivity (the case often used in experiments)
the expressions (2) simplify to
r =
1 − Nt
1 + Nt
. (4)
The latter formula holds for a semi infinite sample, i.e., a sample that is either thicker then the
penetration depth or with roughened back side so that the back side reflected wave does not reach
the detector. In reflection experiment, we measure the ratio of intensities of radiation R = |r|2
. The
intensity of the incident radiation is measured using a reference sample. In infrared range, a golden
coated sample is typically used since gold has an almost spectrally independent reflectivity of about
0.98%.
In case of an anisotropic medium with orthorhombic symmetry, the dielectric function is a tensor
with three diagonal components εx, εy and εz corresponding to the dielectric functions along the x, y
and z axis respectively. In this case, the Fresnel reflection coefficients read
rs =
Ni cos θi − N2
ty − N2
i sin2
θi
Ni cos θi + N2
ty − N2
i sin2
θi
, rp =
NtxNtz cos θi − Ni N2
tz − N2
i sin2
θi
NtxNtz cos θi + Ni N2
tz − N2
i sin2
θi
, (5)
where Ntx, Nty, Ntz is the index of refraction along the x, y and z axis respectively. For a thorough
discussion of a propagation of light in the anisotropic media, see, e.g., Ref. [1].
1.3 Models of the dielectric function
We shell consider several models of the dielectric function that might be usefull in analysis of data
since
• they help to get an insight into the processes responsible for the polarization
• they are Kramers-Kronig consistent, which might not be necessarily true for the data obtained
point by point. Analysis using Kramers-Kronig consistent dielectric function may allow to
determine other unknown parameters of the system, e.g., thicknesses of the layers (including
surface roughness), more components of the dielectric tensor of anisotropic material etc.
• they represent the point-by-point data by small set of parameters.
Below we summarize most frequent models. Since the dielectric function is additive, any combination
of these models can be considered.
1.3.1 Lorentz model
The Lorentz model is derived from the response of harmonic oscillators with density n, mass m and
charge Q. The contribution of a single Lorentz oscillator to the dielectric function can be expressed
as
ε(ω) =
ω2
pl
ω2
0 − ω2 − iωγ
, (6)
The Lorentz oscillator has thus three parameters: the center frequency ω0, the plasma frequency ωpl
and broadening γ. It can be shown that ωpl = nQ2/ 0m. Since the model is derived from Newton
equations, it is inherently Kramers-Kronig consistent. The real and imaginary part of the dielectric
function of one Lorentz oscillator is shown in Fig. 1.
3
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0
- 5
0
5
1 0
ε1
ε
w
ωp l
= 1
ω0
= 1
γ= 0 . 1
ε2
Figure 1: The real and imaginary part of a Lorentz oscillator.
In a more general case, there can be several Lorentz oscillators and the total dielectric function is
then given by the sum
ε = ∞ +
k
ω2
pl,k
ω2
k − ω2 − iωγk
. (7)
The latter formula has in addition the real constant ∞, that can be viewed as a contribution of
polarization processes at a higher frequency than the studied range. Indeed, the low frequency limit
of the Lorentz oscillator (6) is a constant. However, this constant is only an approximation. Instead of
a constant, a more accurate approach would be considering a Lorentz oscillator with zero broadening
at a higher frequency than the studied range. This is the so called Sellmeier model discussed in more
detail below.
The Lorentz model (7), despite its simplicity, is already a versatile model that can be used in a
number of situations. It fits remarkably well the response of infrared active lattice vibrations.
1.3.2 Drude model
The response of free charge carriers, so called Drude model, is obtained as the limit of the Lorentz
oscillator for zero restoring force, that is, for ω0 = 0
εDrude(ω) = −
ω2
pl
ω(ω + iγ)
. (8)
The dielectric function ε(ω) = 1 + εDrude(ω) and the corresponding optical conductivity are shown
in Fig. 2(a) and 2(b), respectively. Note that the ε2(ω) diverges at zero frequency whereas the
conductivity is finite and yields the DC conductivity.
This term usually represents well the frequency response of doped semiconductors or metals. In
case there are multiple bands crossing the Fermi level, one Drude term should be used for each band.
In case of correlated metals, e.g. high temperature superconductors, where the free charge carriers are
interacting, the scattering rate 1/γ depends on frequency. This case can be on a phenomenological
level modelled also using multiple Drude terms.
1.3.3 Sellmeier and Cauchy models
Sellmeier and Cauchy models are used to model the well known dispersion of optical constants in the
transparent frequency range. The Selmaier model is obtained by considering Lorentz oscillators (one
4
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0
- 1 0 0
- 5 0
0
5 0
1 0 0
ε1
w
ωp l
= 1
γ= 0 . 1
ε2
a )
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0
0
5 0 0
1 0 0 0
1 5 0 0
b )
σ1
ε1
w
σ2
Figure 2: The real and imaginary part of the Drude model oscillators with values of parameters shown in the
Figure.
or two are usually enough) at frequencies above the transparent region
ε(ω) =
k
ω2
pl,k
ω2
k − ω2
. (9)
In the simplest case with one oscillator, the model as only two parameters: the frequency and the
plasma frequency.
The Cauchy model is a model for index of refraction obtained from an expansion of the Sellmeier
model at zero frequency
N(λ) = A +
B
λ2
+
C
λ4
+ . . . . (10)
Here, the unknowns are the expansion constants A, B and C. Figure 3 shows the real part of the
dielectric function of one Lorentz oscillator located at ω = 1 and the Cauchy model fitted to the
Sellmeier model up to ω = 0.7. Some deviations between the two models can be noticed arising from
the fact that the Cauchy model is only approximative.
0 . 0 0 . 5 1 . 0
0 . 5
1 . 0
1 . 5
2 . 0
2 . 5
3 . 0
S e l l m a i e r
C a u c h y
ε1
w
ωp l
= 1
ω0
= 1
γ= 0
Figure 3: Sellmeier model with one Lorentzian with zero broadening located at ω = 1 (solid line). Cauchy
model (red dashed line) is fitted to the Sellmeier model up to ω = 0.7.
5
1.3.4 Complex Lorentzian model
In the case the phonon modes are interacting (e.g. due to a nonparabolic potential), the phonon
resonances have an asymmetric shape. This can be modelled using a generalized Lorentz oscillator
with a complex plasma frequency [4]
ε(ω) =
ω2
pl + iωωc
ω2
0 − ω2 − iωγ
. (11)
The complex nominator essentially mixes the real and imaginary part of the standard Lorentz oscillator
so that an asymmetric shape is obtained. This model is often used e.g. for modeling of phonon response
in perovskite oxides that often exhibit an asymmetric shape.
Figure 4 displays the complex Lorentzian profile for various values of ωpl and ωc in such a way
that the absolute value of the nominator is approximately constant. In Fig. 4(a), it can be seen that,
as the imaginary part ωc increases, ε2 gradually transforms to the negatively taken ε1 of the Lorentz
oscillator, and vice versa for the ε1, see Fig. 4(b). The opposite trend is seen for negative values of
ωc, see Figs. 4(c) and (d).
0 5 0 1 0 0 1 5 0 2 0 0
- 5
0
5
1 0
ε1
ω
( b )
0 5 0 1 0 0 1 5 0 2 0 0
- 5
0
5
1 0
ε2
ω
1 0 0 0
9 0 2 0
5 0 7 0
0 1 0 0
ωp l
= ωc
=
( a )
0 5 0 1 0 0 1 5 0 2 0 0
- 1 0
- 5
0
5
ε1
ω
( d )
0 5 0 1 0 0 1 5 0 2 0 0
- 5
0
5
1 0
ωc
=ωp l
=
ε2
ω
1 0 0 0
9 0 - 2 0
5 0 - 7 0
0 - 1 0 0
( c )
Figure 4: The complex Lorentzian model for various values of parameters shown in the panels (a) and (c).
1.3.5 Gaussian lineshape
In some cases, the lineshapes can have the Gaussian shape rather then the shape of a Lorenztian
oscillator. For example, the center frequency might have a gaussian distribution due to a disorder.
The ε2 of the Gaussian lineshape reads
ε2(ω) = A ε−(ω−ω0
σ )
2
+ ε−(ω+ω0
σ )
2
. (12)
The two Gaussians centered at ω0 and −ω0 are necessary so that the total dielectric function fulfills the
Kramers-Kronig requirement ε(ω) = ε(−ω)∗
. The imaginary part of the contribution to the dielectric
function is then calculated numerically using the Kramers-Kronig relations.
6
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0
0
5
1 0
G a u s s i a n o s c .
L o r e n t z o s c .
w
A = 1
ω0
= 1
γ= 0 . 1
ε2
a )
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0
- 5
0
5
b )
G a u s s i a n o s c .
L o r e n t z o s c .
w
ε1
Figure 5: The imaginary (a) and real (b) part of the dielectric function of Gaussian lineshape compared to
the Lorentzian lineshape with the same width at half maximum.
Compared to the Lorentzian function, see Fig. 5, the Gaussian lineshape has an exponential (and
thus much faster) fall-off from the center frequency. In more general case, a convolution between
Gauss and Lorentz lineshape, so called Voigt model, can be considered.
1.3.6 Tauc-Lorentz model
Tauc-Lorentz oscillator was developped by Jellison and Modine [5] for modeling of the dielectric
function of amorphous materials. For E > Eg the formula for ε2 reads
ε2(E) =
AE0C(E − Eg)2
(E2 − E2
0 )2 + C2E2
1
E
(13)
Below the bandgap, the absoprtion is zero. Close to the band edge Eg, the absorption follows the
Tauc law formula, ε2(E) ∝ (E − Eg)2
/E2
. The ε1 is calculated using Kramers-Kronig relations and
can be analytically expressed, see Ref. [1, 5]. Figure 6 displays the dielectric function of the Tauc
Lorentz model for selected values of parameters.
1.3.7 Critical point parabolic band
The critical points of interband transitions of crystallines solids are often analyzed with the critical
point parabolic band model fitted usually to the second or third derivative of the dielectric function
in order to suppress the background [6,7]. In this model, the contribution of a parabolic critical point
(CP) located at an energy ECP to the j-th derivative of the dielectric function reads
dj
ε(E)
dEj
= Aeiφ
(E − ECP + iΓ)−n−j
, (14)
where A and φ are amplitude and phase factor and n has the values -1/2, 0 and 1/2 for three-, twoand
one-dimensional CP, respectively.
1.3.8 Bruggeman effective medium approximation and surface roughness
Effective medium approximations describe the effective optical properties of a composite material
formed by a mixture of two or more components that are assumed to be homogeneously intermixed
7
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0
- 1 0
0
1 0
2 0
ε1
ε
E
A = 1 0 0
E g
= 1
E 0
= 2
C = 1
ε2
Figure 6: The real and imaginary part of the Tauc-Lorentz model for the values of parameters shown in the
figure.
on a scale that is much smaller than the used wavelength of light. There are various approaches how
to calculate the effective dielectric function. One of them is the Bruggeman formula that for circular
inclusions in three dimensions reads
i
δi
εi − εe
εi + 2εe
= 0 . (15)
The summation is effectuated over components indexed with i, with volume fractions δi and dielectric
function εi. The effective dielectric function εe is obtained after solving the latter equation.
The Brugemman formula is used in order to calculate the influence of the surface roughness to
optical measurements. Ellipsometry is particularly sensitive to the surface roughness. A roughness
with magnitude of only few Angstroms gives rise to already noticeable effect on the ellipsometric
angle ∆. In order to take this effect in to account, it is assumed that the surface roughness can be
approximated by a layer composed of 50% by the material and of 50% by air or vacuum. Assuming
a dielectric function of the material, the only free parameter to be determined is the thickness of
the surface roughness layer. In the case there is a transparent region within the measurement range,
the Sellmeier (or Cauchy) model can be used and the thickness of the surface roughness layer is
determined together with the Sellmeier model parameters. In the case there is no absorbing region in
the measurements range, e.g., in metals, this approach cannot be used. One possibility is to determine
the surface roughness magnitude from the atomic force microscopy.
The interfacial roughness between two materials can be modelled in an analogous way.
1.4 Principles of ellipsometry
The ellipsometric measurements make use of the change of the polarization state of light after the
reflection (or transmission) from sample. The simplest ellipsometric configuration with two polarizers
is schematically depicted in Fig. 7. Light coming from the source is polarized by the first polarizer
and is incident on the sample at an angle θi. Since the Fresnel reflection coefficient for s and p
polarized component of the wave have generally different real and imaginary part, the amplitude
and phase of the reflected s and p components differ. They interfere coherently and give rise to an
elliptically polarized state which is analyzed by the second polarizer, the so-called analyzer. Typically,
the intensity of light is measured as a function of the analyzer angle. This is the so-called rotating
analyzer ellipsometry.
In the ellipsometric measurement, the obtained measured quantities are the ellipsometric angles
8
sample
qi
s
p
p
s
~D
Polarizer
Source
Es
Ep
Analyzer
Detector
Figure 7: Scheme depicting the principle of ellipsometry.
Ψ and ∆ defined using the complex quantity ρ as
ρ(Ψ, ∆) = tg Ψei∆
=
rp
rs
. (16)
It can be shown, that on a semi-infinite isotropic sample, the dielectric function of the sample εt can
be expressed as a function of ρ as
εt(Ψ, ∆) = εi sin2
θi 1 + tg 2
θi
1 − ρ(Ψ, ∆)
1 + ρ(Ψ, ∆)
. (17)
The complex dielectric function εt with two unknown quantities (the real and the imaginary part) is
obtained from two measured quantities, Ψ and ∆. In this most simple case, this relation is analytical.
In more complicated cases (e.g. thin films on a substrate), the dielectric function cannot be obtained
analytically and has to be obtained numerically using regression analysis, where the optical constants
are sought in order that the theoretically calculated Ψ and ∆ matches the experimentally measured
values. However, even in the case of thin films, it is often instructive to look at the data using
Eq. (17) since the resulting εt already does not depend on the angle of incidence in the trivial way as
the ellipsometric angles Ψ and ∆ do. However in this case it cannot be considered as the dielectric
function, so it is called the pseudo-dielectric function.
In the rotating analyzer ellipsometry, the quantities that are determined are actually tg Ψ and
cos ∆. Therefore the errorbar of ∆ diverges as ∆ approaches 0 and 180◦
. Mainly for this reason,
advanced ellipsometric setups were developed that employ waveplates (so called compensators) that
shift the phase betwen s and p polarized wave. A series of measuremets with different degrees of phase
shifts are measured in the so-called rotating compensator ellipsometry. As a results ∆ is precisely
measured over the whole range 0-360◦
. In addition, depolarization can be determined. Depolarization
arises when different polarization states interfere incoherently. Typically, it arises due to a finite
frequency resolution of monochromator, lateral inhomogeneities either of the dielectric function or
of film thickness; or due to reflections from the back side of the sample. These situations can be
in principle modelled using Stokes vectors and Muller matrices that can handle partially polarized
light [2].
1.4.1 Available spectroscopic equippment
The practicum tasks will be measured at nano core facility at CEITEC using Woollam VASE ellipsometer
for near to ultraviolet range (0.5 to 6.5 eV) and Woollam IR-VASE ellipsometer for mid
infrared range (300 to 6000 cm−1
). The far-infrared range (50 to 700 cm−1
) can be covered with
near normal incidence reflection and transmission measurements using the vacuum Fourier transform
spectrometer Bruker Vertex 70V.
9
The VASE ellipsometer is equipped with a double grating monochromator, photoconductive detector
(Si/InGaAs) and a photomultiplier for high sensitivity in the UV range. It has a rotating
compensator equivalent technology, the so called auto-retarder. The IR-VASE ellipsometer is a
Fourier-transform spectrometer based ellipsometer equipped with a rotating compensator. Since both
ellipsometers have rotating compensator, they can determine depolarization and ∆ in the whole range
0-360◦
. In principle, they can measure 11 elements of the Muller matrix.
For analysis of the obtained data, the Woollam WVASE software can be employed.
1.5 Practicum tasks
1.5.1 Phonon spectra of isotropic and anisotropic semiconductors
1. Measure optical response of SrTiO3 (isotropic crystal), GaN and SiO2 (anisotropic crystals)
in the frequency range of infrared active phonons. Use the polarized far-infrared reflection
measurements complemented in mid-infrared range by ellipsometry.
2. Model the reflectivity and ellipsometry spectra by appropriate models and display the obtained
dielectric function, and phonon parameter values.
1.5.2 Direct interband transitions in semiconductors
1. Measure the ellipsometric spectra of Si, GaN, CdS and InAs semiconductors.
2. Perform second derivative of the obtained pseudo-dielectric function and estimate the energy of
the critical points.
3. Correct the data for the surface roughness and obtain a point-by point dielectric function of the
materials.
4. Model the dielectric function of the materials with a suitable Kramers-Kronig consistent model.
1.5.3 Free charge carrier response in semiconductors and metals
1. Measure the ellispometric spectra of gold, doped Si, and a thin film of La0.7Sr0.3CoO3 grown on
LSAT substrate with VASE and IR-VASE ellipsometers.
2. Model the obtained spectra with appropriate models and determine the parameters of the Drude
term. Estimate the concentration of charge carriers assuming tabulated value of the effective
mass. Extrapolate the conductivity to zero frequency and compare it with tabulated DC conductivity
values.
References
[1] H. Fujiwara, Spectroscopic Ellipsometry, Principles and applications, John Wiley & Sons Ltd,
Chichester, 2007.
[2] Handbook of Ellipsometry, Tompkins, G. and Irene, E. edts. , A. William Andrew Publishing and
Springer-Verlag GmbH & Co. KG, 2005.
[3] R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, North Holland; 3rd reprint
1999 edition, 1999.
[4] J. Huml´ıˇcek, R. Henn, and M. Cardona, Phys. Rev. B 61, 14 554 (2000).
[5] G. E. Jellison, Jr. and F. A. Modine, Appl. Phys. Lett. 69, 371 (1996), Erratum, Appl. Phys.
Lett. 69, 2137 (1996).
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[6] M. Cardona, Solid state physics, Modulation Spectroscopy, 1969.
[7] J. Huml´ıˇcek and M. Garriga, Temperature Dependence of the Optical Spectra of Si1−xGex Alloys,
Silicon-Germanium Carbon Alloys Growth, Properties and Applications 483, 2002.